Simplify and expand the following expression: $ \dfrac{4t - 4}{3t - 7}+\dfrac{4t}{4t + 9} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(3t - 7)(4t + 9)$ Multiply the first term by $\dfrac{4t + 9}{4t + 9}$ $ \begin{align*} \dfrac{4t - 4}{3t - 7} \times \dfrac{4t + 9}{4t + 9} & = \dfrac{(4t - 4)(4t + 9)}{(3t - 7)(4t + 9)} \\ & = \dfrac{16t^2 + 20t - 36}{(3t - 7)(4t + 9)}\end{align*} $ Multiply the second term by $\dfrac{3t - 7}{3t - 7}$ $ \begin{align*} \dfrac{4t}{4t + 9} \times \dfrac{3t - 7}{3t - 7} & = \dfrac{(4t)(3t - 7)}{(4t + 9)(3t - 7)} \\ & = \dfrac{12t^2 - 28t}{(4t + 9)(3t - 7)}\end{align*} $ Now we have: $ = \dfrac{16t^2 + 20t - 36}{(3t - 7)(4t + 9)} + \dfrac{12t^2 - 28t}{(4t + 9)(3t - 7)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{16t^2 + 20t - 36 + 12t^2 - 28t}{(3t - 7)(4t + 9)} $ $ = \dfrac{28t^2 - 8t - 36}{(3t - 7)(4t + 9)}$ Expand the denominator: $ = \dfrac{28t^2 - 8t - 36}{12t^2 - t - 63}$